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A290943 Number of ways to write n as an ordered sum of 3 generalized pentagonal numbers (A001318).

%I A290943 #24 Jun 27 2025 22:23:10
%S A290943 1,3,6,7,6,6,7,12,12,12,9,6,12,12,18,13,12,18,12,18,12,13,18,12,24,12,
%T A290943 12,24,21,30,12,18,18,12,24,18,19,18,24,24,18,24,36,24,18,19,18,24,24,
%U A290943 30,18,12,36,30,24,21,18,36,24,36,24,12,36,36,36,18,25,30,24,24,24,30,24,36,30,24
%N A290943 Number of ways to write n as an ordered sum of 3 generalized pentagonal numbers (A001318).
%C A290943 Conjecture: every number is the sum of at most k - 4 generalized k-gonal numbers (for k >= 8).
%C A290943 In 1830, Legendre showed that for each integer m>4 every integer N >= 28*(m-2)^3 can be written as the sum of five m-gonal numbers. In 1994 R. K. Guy proved that each natural number is the sum of three generalized pentagonal numbers. In a 2016 paper Zhi-Wei Sun proved that each natural number is the sum of four octagonal numbers. - _Zhi-Wei Sun_, Oct 03 2020
%H A290943 Robert Israel, <a href="/A290943/b290943.txt">Table of n, a(n) for n = 0..10000</a>
%H A290943 Ilya Gutkovskiy, <a href="/A290943/a290943.pdf">Extended graphical example</a>
%H A290943 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2015.10.014">A result similar to Lagrange's theorem</a>, J. Number Theory 162(2016), 190-211.
%H A290943 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal Number</a>
%H A290943 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F A290943 G.f.: (Sum_{k=-infinity..infinity} x^(k*(3*k-1)/2))^3.
%F A290943 G.f.: (Sum_{k>=0} x^A001318(k))^3.
%F A290943 G.f.: Product_{n >= 1} ( (1 - q^(3*n))/(1 - q^n + q^(2*n)) )^3. - _Peter Bala_, Jan 04 2025
%e A290943 a(6) = 7 because we have [5, 1, 0], [5, 0, 1], [2, 2, 2], [1, 5, 0], [1, 0, 5], [0, 5, 1] and [0, 1, 5].
%p A290943 N:= 100;
%p A290943 bds:= [fsolve(k*(3*k-1)/2 = N)];
%p A290943 G:= add(x^(k*(3*k-1)/2),k=floor(min(bds))..ceil(max(bds)))^3:
%p A290943 seq(coeff(G,x,n),n=0..N); # _Robert Israel_, Aug 16 2017
%t A290943 nmax = 75; CoefficientList[Series[Sum[x^(k (3 k - 1)/2), {k, -nmax, nmax}]^3, {x, 0, nmax}], x]
%t A290943 nmax = 75; CoefficientList[Series[Sum[x^((6 k^2 + 6 k + (-1)^(k + 1) (2 k + 1) + 1)/16), {k, 0, nmax}]^3, {x, 0, nmax}], x]
%t A290943 nmax = 75; CoefficientList[Series[EllipticTheta[4, 0, x^3]^3/QPochhammer[x, x^2]^3, {x, 0, nmax}], x]
%Y A290943 Cf. A001318, A002175, A008443, A080995, A093518, A093519, A255350, A255934, A256132, A256171, A280718.
%K A290943 nonn,easy
%O A290943 0,2
%A A290943 _Ilya Gutkovskiy_, Aug 14 2017